Sausage, English Muffin, and Cream Gravy Sandwich

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10 large eggs

2 tablespoons cold water

6 green onions, white and green parts, thinly sliced, plus chopped green onion tops, for garnish

Salt and freshly ground black pepper

4 tablespoons unsalted butter

4 thick cut English muffins, toasted

Sausage and Gravy, recipe follows

1 cup grated Monterey jack or white Cheddar cheese

Sausage and Gravy:

1 pound breakfast sausage, casings removed

1/2 large onion or 1 small onion, finely chopped

1 jalapeno chile, finely chopped

2 tablespoons all-purpose flour

1 1/2 cups whole milk

1 tablespoon finely chopped fresh sage leaves

Salt and freshly ground black pepper